Simple approximations of semialgebraic sets and their applications to control
Simple approximations of semialgebraic sets and their applications to control
复制标题
DOI：
10.1016/j.automatica.2016.11.021
复制
发表时间：
2017-04-01
期刊：
影响因子：
6.4
通讯作者：
Lagoa, Constantino M.
中科院分区：
文献类型：
--
作者：
Dabbene, Fabrizio;Henrion, Didier;Lagoa, Constantino M.
在控制系统分析和设计中遇到的许多不确定性集合都可以用半代数集合表示，即用多项式不等式描述的集合的交集。重要的例子是例如舒尔和赫尔维茨稳定域。这些集合通常具有非常复杂的形状（非凸的，甚至是非连通的），这使得它们的操作变得困难。因此，找到这些集合的足够简单的近似值是相当重要的，能够捕捉它们的主要特征，同时保持低水平的复杂性。由于这些原因，在过去的几年中，一些凸近似，例如基于超矩形，多面体，或椭球已经proposed.In这项工作中，我们进一步提出了可能的非凸的，但仍然简单的近似，基于一个小体积的多项式superlevel集的一个单一的正多项式的次数。我们展示了如何这些集合可以很容易地近似通过最小化的L-1范数的多项式的半代数集，受正性约束。直觉上，这对应于最小体积椭球问题中常见的迹最小化启发式。从计算的角度来看，我们设计了一个层次的凸线性矩阵不等式问题，以产生这些近似，我们提供了理论上严格的收敛结果，在这个意义上说，层次的外近似收敛的体积（或者，等价地，几乎处处且几乎一致地）到原始集合。最后，我们展示了如何多项式超水平集的概念可以用来生成均匀分布在给定的半代数集上的样本。不同的数值例子表明所提出的方法的效率。(C)2016爱思唯尔有限公司版权所有
Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the Schur and Hurwitz stability domains. These sets often have very complicated shapes (nonconvex, and even non-connected), which render difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrectangles, polytopes, or ellipsoids have been proposed.In this work, we move a step further, and propose possibly nonconvex yet still simple approximations, based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L-1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encountered in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of convex linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set.Finally, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples. (C) 2016 Elsevier Ltd. All rights reserved.